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Office of Energy Efficiency & Renewable Energy

Operated by the Alliance for Sustainable Energy, LLC

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Contract No. DE-AC36-08GO28308

Definition of the 

Semisubmersible Floating 

System for Phase II of OC4

A. Robertson, J. Jonkman, M. Masciola, 

and H. Song

National Renewable Energy Laboratory

A. Goupee and A. Coulling

University of Maine

C. Luan

Norwegian University of Science

Technical Report

NREL/TP-5000-60601

September 2014
NREL is a national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

Operated by the Alliance for Sustainable Energy, LLC

This report is available at no cost from the National Renewable Energy

Laboratory (NREL) at www.nrel.gov/publications.

Contract No. DE-AC36-08GO28308

National Renewable Energy Laboratory

15013 Denver West Parkway

Golden, CO 80401

303-275-3000 • www.nrel.gov

Definition of the 

Semisubmersible Floating 

System for Phase II of OC4

A. Robertson, J. Jonkman, M. Masciola, 

and H. Song

National Renewable Energy Laboratory

A. Goupee and A. Coulling

University of Maine

C. Luan

Norwegian University of Science

Prepared under Task No. WE14.5A01

Technical Report

NREL/TP-5000-60601

September 2014
NOTICE

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Cover Photos: (left to right) photo by Pat Corkery, NREL 16416, photo from SunEdison, NREL 17423, photo by Pat Corkery, NREL 

16560, photo by Dennis Schroeder, NREL 17613, photo by Dean Armstrong, NREL 17436, photo by Pat Corkery, NREL 17721.

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iii

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Table of Contents

List of Figures ............................................................................................................................................ iv

List of Tables .............................................................................................................................................. iv

1 Introduction........................................................................................................................................... 1

2 Tower Properties .................................................................................................................................. 3

3 Floating Platform Structural Properties ............................................................................................. 5

3.1 General Properties............................................................................................................................ 5

3.2 Flexible Properties........................................................................................................................... 9

3.3 Coordinate System......................................................................................................................... 10

4 Floating Platform Hydrodynamic Properties................................................................................... 11

4.1 Overview........................................................................................................................................ 11

4.2 Hydrostatics................................................................................................................................... 11

4.3 Hydrodynamics.............................................................................................................................. 12

4.3.1 Dimensionless Numbers for Flow/Structure Interaction................................................. 12

4.3.2 Potential-Flow Theory .................................................................................................... 16

4.3.3 Morison’s Equation......................................................................................................... 20

4.3.4 Quadratic Drag for Other Models................................................................................... 24

4.4 Summary........................................................................................................................................ 25

5 Mooring System Properties............................................................................................................... 26

5.1 Overview........................................................................................................................................ 26

5.2 Linearized Mooring Model............................................................................................................ 27

5.3 Nonlinear Mooring Model ............................................................................................................. 28

5.4 Nonlinear Mooring Model with One Line ..................................................................................... 31

6 Control System Properties ................................................................................................................ 34

References................................................................................................................................................. 35

Appendix I: Calculation of Average Cd Value....................................................................................... 37
iv

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List of Figures

Figure 1-1: DeepCwind floating wind system design.............................................................................. 1

Figure 3-1: As-built picture of DeepCwind semisubmersible for 1/50th scale tests............................ 5

Figure 3-2: Plan (left) and side (right) view of the DeepCwind semisubmersible platform 

(abbreviations can be found in Table 3-2) ......................................................................................... 6

Figure 3-3: Side view of platform walls and caps (abbreviations can be found in Table 3-2) ............ 6

Figure 4-1: Dimensionless parameters for the OC4 DeepCwind semisubmersible.......................... 15

Figure 4-2: Hydrodynamic wave excitation per unit amplitude for the OC4 semisubmersible 

for 0° wave heading........................................................................................................................... 17

Figure 4-3: Hydrodynamic added mass and damping for the OC4-DeepCwind semisubmersible.. 18

Figure 4-4: Simplified plan footprint of the OC4 semisubmersible ..................................................... 19

Figure 4-5: Radiation impulse-response functions for the OC4-DeepCwind semisubmersible....... 20

Figure 4-6: Distribution of the drag coefficient as a function of the Re number............................... 22

Figure 5-1: Mooring line arrangement .................................................................................................... 27

Figure 5-2: Load-displacement relationships for the OC3-Hywind mooring system in 1D.............. 30

Figure 5-3: Load-displacement relationships for the OC4 DeepCwind semisubmersible mooring 

system in 2D........................................................................................................................................ 31

Figure 5-4: Mooring line in a local coordinate system ......................................................................... 32

Figure 5-5: Load-displacement relationships for one mooring line .................................................... 33

Figure AI-1: Variation of the drag coefficient across all sea-states. The mean value for Cd at 

incremental depths is also registered in the figures. ..................................................................... 38

List of Tables

Table 2-1: Distributed Tower Properties.................................................................................................. 4

Table 2-2: Undistributed Tower Properties ............................................................................................. 4

Table 3-1: Floating Platform Geometry..................................................................................................... 7

Table 3-2: Member Geometry..................................................................................................................... 7

Table 3-3: Floating Platform Structural Properties.................................................................................. 9

Table 3-4: Properties of Members ........................................................................................................... 10

Table 4-1: Applicable Sections for Hydrodynamics Modeling Approaches ...................................... 11

Table 4-2: Periodic Sea State Definitions .............................................................................................. 14

Table 4-3: Ratio of Diameter/Wave Length............................................................................................ 15

Table 4-4: Mean Cd Values Across All Sea States for the Full-Scale OC4 Semisubmersible ........... 22

Table 4-5: Quadratic Drag Coefficients for the FAST Model ................................................................ 24

Table 4-6: Floating Platform Hydrodynamic Properties........................................................................ 25

Table 5-1: Mooring System Properties ................................................................................................... 26

Table 6-1: Baseline Control System Property Modifications............................................................... 34
1

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1 Introduction

Phase II of the Offshore Code Comparison Collaboration Continuation (OC4) project involved 

modeling of a semisubmersible floating offshore wind system as shown below in Figure 1-1.

This report documents the specifications of the floating system, which were needed by the OC4 

participants for building aero-hydro-servo-elastic models.

Figure 1-1: DeepCwind floating wind system design

OC4 is a continuation of the OC3 project (Offshore Code Comparison Collaboration) that 

examined three different fixed-bottom and one floating offshore wind systems. The floating 

concept analyzed for OC3 was the OC3-Hywind system (Jonkman 2010), a spar concept. For 

77.6 m

SWL

Top of tower

6 m

20 m

26 m

Tower freeboard

10 m
2

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OC4, a semisubmersible design developed for the DeepCwind project is used. DeepCwind is a 

U.S.-based project aimed at generating test data for use in validating floating offshore wind 

turbine modeling tools. In the first phase of this project, three different floating wind turbine 

configurations were tested at 1/50th scale in a wave basin under combined wind/wave loading 

(Goupee, et al 2012). The three configurations tested are considered to span the design space of 

floating wind turbine concepts. They included a spar similar to the OC3-Hywind, a tension-leg 

platform, and a semisubmersible. Analyzing the DeepCwind semisubmersible design in the OC4 

project (at full scale) creates the opportunity for a follow-on project related to validation of the 

simulated dynamics of the system by comparison to the wave-tank data. The focus of OC4, 

however, was verification of modeling tools by comparing results of simulated responses 

between various tools. 

The semisubmersible substructure design examined in this phase is based on the as-built 

configuration used in the DeepCwind tests. The as-built tower and turbine from these tests, 

however, were not used. Instead, the turbine specification for OC4 Phase II was the National 

Renewable Energy Laboratory (NREL) offshore 5-MW baseline wind turbine (Jonkman, et al 

2009), which is a representative utility-scale, multimegawatt turbine. This turbine was used in 

the OC3 project and was used in all phases of OC4 as well. The tower supporting the wind 

turbine changes slightly between the different phases, depending on the design of the 

substructure (jacket, semisubmersible) chosen. The control system properties also change to 

accommodate the differences in system dynamics. 

This report presents the data needed to support modeling activities for Phase II of the OC4 

project. The material is summarized in the following sections:

• Section 2: Tower Properties

• Section 3: Floating Platform Structural Properties

• Section 4: Floating Platform Hydrodynamic Properties

• Section 5: Mooring System Properties

• Section 6: Control System Properties
3

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2 Tower Properties

The tower used for the OC4-DeepCwind semisubmersible is nearly identical to the OC3-Hywind 

spar-buoy tower. The only difference is the tower mode shapes, which are used by some tools to 

represent the flexibility of the tower. The mode shapes differ due to the change in boundary 

conditions at the support of the tower resulting from changes in the support platform and 

mooring. This section summarizes the general tower properties and is identical to the information 

provided in the floating system definition document for OC3 Phase IV (Jonkman 2010).

The base of the tower is coincident with the top of the main column of the semisubmersible and 

is located at an elevation of 10 m above the still water level (SWL). The top of the tower is 

coincident with the yaw bearing and is located at an elevation of 87.6 m above the SWL. This 

tower-top elevation—and the corresponding 90-m elevation of the hub above the SWL—is 

consistent with the land-based version of the NREL 5-MW baseline wind turbine (as given in 

[Jonkman et al, 2009]). These properties are all relative to the undisplaced position of the 

platform. 

The distributed properties of the tower for the NREL 5-MW baseline wind turbine atop the OC4 

DeepCwind semisubmersible are founded on the base diameter of 6.5 m, which matches the 

diameter of the main column of the semisubmersible (see Section 3), and the tower-base 

thickness (0.027 m), top diameter (3.87 m) and thickness (0.019 m), and effective mechanical 

steel properties of the tower used in the DOWEC study (as given in Table 9 on page 31 of 

[Kooijman, et al 2003]). The Young’s modulus was taken to be 210 GPa, the shear modulus was 

taken to be 80.8 GPa, and the effective density of the steel was taken to be 8,500 kg/m3

. The 

density of 8,500 kg/m3 was meant to be an increase above steel’s typical value of 7,850 kg/m3 to 

account for paint, bolts, welds, and flanges that are not included in the tower thickness data. 

This value is only used for the density of steel in the tower and not elsewhere in the structure.

The radius and thickness of the tower are assumed to be linearly tapered from the tower base to 

tower top. Table 2-1 gives the resulting distributed tower properties.

The entries in the first column, “Elev,” are the vertical locations along the tower centerline 

relative to the SWL. “HtFr” is the fractional height along the tower centerline from the tower 

base (0.0) to the tower top (1.0). The rest of the columns are similar to those described for the 

distributed blade properties presented in (Jonkman, et al 2009). There is no offset in the center of 

gravity for the individual sections.

The resulting overall (integrated) tower mass is 249,718 kg and is centered (i.e. the center of 

mass [CM]) of the tower, is located] at 43.4 m along the tower centerline above the SWL. This is 

derived from the overall tower length of 77.6 m. 

A structural-damping ratio of 1% critical is specified for all modes of the isolated tower 

(cantilevered atop a rigid foundation without the rotor-nacelle assembly mass present), which 

corresponds to the values used in the DOWEC study (from page 21 of [Kooijman, et al 2003]).
4

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Table 2-1: Distributed Tower Properties

Elev

(m)

HtFr

(-)

TMassDen 

(kg/m)

TwFAStif 

(N-m2)

TwSSStif 

(N-m2)

TwGJStif 

(N-m2)

TWEAStif 

(N)

TwFAIner 

(kg-m)

TwSSIner 

(kg-m)

10.0 0.0 4667.00 603.903E+9 603.903E+9 464.718E+9 115.302E+9 24443.7 24443.7

17.76 0.1 4345.48 517.644E+9 517.644E+9 398.339E+9 107.354E+9 20952.2 20952.2

25.52 0.2 4034.76 440.925E+9 440.925E+9 339.303E+9 99.682E+9 17847.0 17847.0

33.28 0.3 3735.44 373.022E+9 373.022E+9 287.049E+9 92.287E+9 15098.5 15098.5

41.04 0.4 3447.32 313.236E+9 313.236E+9 241.043E+9 85.169E+9 12678.6 12678.6

48.80 0.5 3170.40 260.897E+9 260.897E+9 200.767E+9 78.328E+9 10560.1 10560.1

56.56 0.6 2904.69 245.365E+9 245.365E+9 165.729E+9 71.763E+9 8717.2 8717.2

64.32 0.7 2650.18 176.028E+9 176.028E+9 135.458E+9 65.475E+9 7124.9 7124.9

72.08 0.8 2406.88 142.301E+9 142.301E+9 109.504E+9 59.464E+9 5759.8 5759.8

79.84 0.9 2174.77 113.630E+9 113.630E+9 87.441E+9 53.730E+9 4599.3 4599.3

87.60 1.0 1953.87 89.488E+9 89.488E+9 68.863E+9 48.272E+9 3622.1 3622.1

Elev = Elevation

HtFr = Height Fraction

TMassDen = Tower Mass Density

TwFAStif = Tower Fore/Aft Stiffness

TwSSStif = Tower Side/Side Stiffness

TwGJStif = Tower GJ Stiffness

TWEAStif = Tower EA Stiffness

TwFAIner = Tower Fore/Aft Inertia

TwSSIner = Tower Side/Side Inertia

Table 2-2 summarizes the undistributed tower properties discussed in this section.

Table 2-2: Undistributed Tower Properties

Elevation to Tower Base (Platform Top) Above SWL 10 m

Elevation to Tower Top (Yaw Bearing) Above SWL 87.6 m

Overall (Integrated) Tower Mass 249,718 kg

CM Location of Tower Above SWL Along Tower Centerline 43.4 m

Tower Structural-Damping Ratio (All Modes) 1%
5 

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3 Floating Platform Structural Properties

3.1 General Properties

Details of the floating platform structure are provided in Figure 3-1 and Figure 3-2 (an 

explanation of the abbreviations for the members is provided in Table 3-2). The draft of the 

platform is 20 m. The tower is cantilevered at an elevation of 10 m above the SWL to the top of 

the main column (MC) of the floating platform. Between the top and bottom of the platform, the 

OC4 DeepCwind semisubmersible consists of a main column attached to the tower and three 

offset columns that are connected to the main column through a series of smaller diameter 

pontoons and cross members. There are five sets of these smaller members (members in black in 

Figure 3-1): 

• Two sets of three pontoons (for a total of six members) connecting the offset columns 

form a triangle (or delta), at the top (DU 1-3) and bottom (DL 1-3) of the 

semisubmersible

• Two sets of three pontoons (for a total of six members) connecting the offset columns 

with the main column form a y-connection at the top (YU 1-3) and bottom (YL 1-3) of 

the semisubmersible 

• Three cross braces connecting the bottom of the main column with the top of the offset 

columns 

Figure 3-: As-built picture of DeepCwind semisubmersible for 1/50th scale tests

Each offset column (UC 1-3) starts above the SWL and continues beneath the water. At the base 

of the three offset columns is a larger diameter cylinder, or base column (BC 1-3), which helps to 

suppress motion (particularly in the heave direction, but also in surge, sway, roll, and pitch). A 

summary of the geometry, including the diameters of each of the members is given in Table 3-1

and Table 3-2. These properties are all relative to the undisplaced position of the platform. The 

walls and boundaries of the caps are shown in Figure 3-3. 
6 

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Figure 3-: Plan (left) and side (right) view of the DeepCwind semisubmersible platform

(abbreviations can be found in Table 3-2) 

Figure 3-: Side view of platform walls and caps (abbreviations can be found in Table 3-2) 

40.87 m Y

X

DL1, DU1

UC1 BC1

BC2 UC2

BC3 UC3

DL2, DU2

DL3, DU3

YL2, 

YU2,

CB2

YL1, 

YU1,

CB1

YL3, 

YU3,

CB3

MC

Y

Z

UC3 UC1

BC3 BC1

CB3 CB1

DL3

DU3

MC
7 

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 Table 3-: Floating Platform Geometry

Depth of platform base below SWL (total draft) 20 m

Elevation of main column (tower base) above SWL 10 m

Elevation of offset columns above SWL 12 m 

Spacing between offset columns 50 m

Length of upper columns 26 m

Length of base columns 6 m

Depth to top of base columns below SWL 14 m

Diameter of main column 6.5 m 

Diameter of offset (upper) columns 12 m

Diameter of base columns 24 m

Diameter of pontoons and cross braces 1.6 m

Table 3-: Member Geometry

Column Name Abbr. Start location 

(X,Y,Z) 

End location

(X,Y,Z) 

Length 

(m)

Wall 

Thick. 

(m)

Main Column MC (0, 0,-20) (0, 0,10) 30 0.03

Upper Column 1 UC1 (14.43, 25, -14) (14.43, 25, 12) 26 0.06

Upper Column 2 UC2 (-28.87, 0, -14) (-28.87, 0, 12) 26 0.06

Upper Column 3 UC3 (14.43, -25, -14) (14.43, -25, 12) 26 0.06

Base Column 1 BC1 (14.43, 25, -20) (14.43, 25, -14) 6 0.06

Base Column 2 BC2 (-28.87, 0, -20) (-28.87, 0, -14) 6 0.06

Base Column 3 BC3 (14.43, -25, -20) (14.43, -25, -14) 6 0.06

Delta Pontoon, 

Upper 1 DU1 (9.20, 22, 10) (-23.67, 3, 10) 38 0.0175

Delta Pontoon, 

Upper 2 DU2 (-23.67, -3, 10) (9.20, -22, 10) 38 0.0175

Delta Pontoon, 

Upper 3 DU3 (14.43, -19, 10) (14.43, 19, 10) 38 0.0175

Delta Pontoon, 

Lower 1 DL1 (4, 19, -17) (-18.47, 6, -17) 26 0.0175

Delta Pontoon, 

Lower 2 DL2 (-18.47, -6, -17) (4, -19, -17) 26 0.0175

Delta Pontoon, 

Lower 3 DL3 (14.43, -13, -17) (14.43, 13, -17) 26 0.0175
8

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Y Pontoon, Upper 1 YU1 (1.625, 2.815, 10) (11.43, 19.81, 10) 19.62 0.0175

Y Pontoon, Upper 2 YU2 (-3.25, 0, 10) (-22.87, 0, 10) 19.62 0.0175

Y Pontoon, Upper 3 YU3 (1.625, -2.815, 10) (11.43, -19.81, 10) 19.62 0.0175

Y Pontoon, Lower 1 YL1 (1.625, 2.815, -17) (8.4, 14.6, -17) 13.62 0.0175

Y Pontoon, Lower 2 YL2 (-3.25, 0, -17) (-16.87, 0, -17) 13.62 0.0175

Y Pontoon, Lower 3 YL3 (1.625, -2.815, -17) (8.4, -14.6, -17) 13.62 0.0175

Cross Brace 1 CB1 (1.625, 2.815, -16.2) (11.43, 19.81, 9.13) 32.04 0.0175

Cross Brace 2 CB2 (-3.25, 0, -16.2) (-22.87, 0, 9.13) 32.04 0.0175

Cross Brace 3 CB3 (1.625, -2.815, -16.2) (11.43, -19.81, 9.13) 32.04 0.0175

Upper Col. Top Cap UCTC 0.06

Upper Col. Bottom 

Cap UCBC 0.06

Base Col. Top Cap BCTC 0.06

Base Col. Bottom 

Cap BCBC 0.06

Main Col. bottom 

cap

MCBC 0.03

The mass, including ballast, of the floating platform is 1.3473E+7 kg. This mass was calculated 

such that the combined weight of the rotor-nacelle assembly, tower, and floating platform, plus 

the weight of the mooring system (not including the small portion resting on the seafloor) in 

water balances with the buoyancy (i.e. weight of the displaced fluid) of the undisplaced platform 

in still water. The CM of the floating platform, which includes everything except the tower, 

rotor-nacelle assembly, and moorings, is located at 13.46 m along the platform centerline below 

the SWL. This value was set so that the overall system CM matched that of the scale-model 

DeepCwind system. The roll and pitch inertias of the floating platform about its CM are both 

6.827E+9 kg-m

2 about the platform x-axis and y-axis respectively, and the yaw inertia of the 

floating platform about its centerline is 1.226E+10 kg-m2

. 

Table 3-3 summarizes the undisplaced platform properties discussed in this section.
9 

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Table 3-: Floating Platform Structural Properties

Platform mass, including ballast 1.3473E+7 kg

CM location below SWL 13.46 m 

Platform roll inertia about CM 6.827E+9 kg-m2

Platform pitch inertia about CM 6.827E+9 kg-m2

Platform yaw inertia about CM 1.226E+10 kg-m2

3.2 Flexible Properties

The flexible properties of the OC4-DeepCwind semisubmersible were derived by the authors to 

achieve a prescribed set of values for the metal mass of the system and water ballasting. The total 

metal mass of the structure is 3.8522E+6 kg, and the total water mass is 9.6208E+6 kg. To 

match these specified values, steel (density = 7850 kg/m^3) was used for all members with wall 

thicknesses as defined in Table 3-2. 

An ANSYS model of the structure with these dimensions was built, and it was found that the 

free-free bending modes of the platform were fairly low, most likely because the pontoon and 

cross members were reasonably sized for a model-scale semisubmersible but too narrow at full 

scale. Based on input from commercial developers, the bending stiffness of the members was 

increased until the lowest free-free bending mode was greater than 3 Hz. This was accomplished 

by increasing the Young’s Modulus of steel for all members from a typical value of 2.10E+2

GPa to 2.10E+4 GPa, an increase of two orders of magnitude. We chose to increase the Young’s 

Modulus, rather than change the member size to achieve a desired bending stiffness, to stay 

consistent with the DeepCwind geometry and weights. For a realistic full-scale system, the size 

of the cross braces and pontoons would need to be increased and internal stiffeners would need 

to be added to the columns. The model-scale DeepCwind system was built using aluminum 

wrapped around a foam core, which provided an increased stiffness of the individual members. 

For those modeling the system as beams, the goal is to have an effective bending stiffness (EI) 

that is comparable to a member with the wall thicknesses and Young’s Modulus described 

above. These EI values are summarized in Table 3-4, as well as the torsional (GJ) and 

extensional stiffness (EA). These stiffness values are calculated based on the cross-sectional

geometry; however, it may not be appropriate to effectively model the base columns as beams 

because of their low aspect ratio. In the calculation of the shear modulus (G), a Poisson’s ratio 

of 0.33 was used. The structural damping ratio for all members is set at a value of 1%. 

Table 3-4 also provides a description of the mass per unit length of each member and the total 

mass for that member. One should note that the end caps (UCTC, UCBC, BCBC, and MCBC) 

are located inside the walls of the cylinders. It is up to the modeler to determine how to 

incorporate these end caps and the ballasting that is described next.

The structure is ballasted with water (density = 1025 kg/m3

) in the offset columns. The water is 

compartmentalized and is not allowed to move or slosh. The three upper columns are filled with 

water from the bottom up to 6.17 m below the SWL, and the base columns are filled from draft 

to 5.1078 m above draft. To achieve a platform CM that matched that of the model-scale 
10 

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DeepCwind system, the base columns were not completely filled. There is no water ballast in 

the pontoons, cross braces, or main column. 

Table 3-: Properties of Members

Member

Name

Metal mass 

(kg) 

Mass per 

unit length

( kg/m) 

EI 

(N-m2

) 

GJ

(N-m2) 

EA 

(N)

MC 1.436E+05 4.787E+03 6.701E+13 5.038E+13 1.281E+13

UC 4.594E+05 1.767E+04 8.423E+14 6.333E+14 4.726E+13

BC 2.125E+05 3.542E+04 6.789E+15 5.105E+15 9.476E+13

DU 2.599E+04 6.830E+02 5.720E+11 4.301E+11 1.827E+12

DL 1.778E+04 6.830E+02 5.720E+11 4.301E+11 1.827E+12

YU 1.329E+04 6.830E+02 5.720E+11 4.301E+11 1.827E+12

YL 9.164E+03 6.830E+02 5.720E+11 4.301E+11 1.827E+12

CB 2.142E+04 6.830E+02 5.720E+11 4.301E+11 1.827E+12

UCTC 5.221E+04 -- -- -- -- 

UCBC 5.327E+04 -- -- -- -- 

BCTC 1.577E+05 -- -- -- -- 

BCBC 2.109E+05 -- -- -- -- 

MCBC 7.671E+03 -- -- -- -- 

3.3 Coordinate System

In the following sections, several of the platform specifications refer to an inertial reference 

frame and platform degrees-of-freedom (DOFs). In these sections, X, Y, and Z represent the set 

of orthogonal axes of the inertial reference frame, with the XY-plane designating the SWL and 

the Z-axis directed upward (opposite gravity) along the centerline of the undisplaced platform.

The X-axis is the nominal wind direction (a wind direction of zero) and the Y-axis is lateral to the 

left when looking downwind. The direction of the X and Y axes in relation to the structure is 

shown in Figure 3-2. The rigid-body platform DOFs include translational surge, sway, and 

heave motions and rotational roll, pitch, and yaw motions. Positive surge is defined along the 

positive X-axis, sway is along the Y-axis, and heave is along the Z-axis. Positive roll is defined 

about the positive X-axis, pitch is about the Y-axis, and yaw is about the Z-axis.
11 

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4 Floating Platform Hydrodynamic Properties

4.1 Overview

Hydrodynamic loads on offshore wind systems include contributions from linear hydrostatics, 

linear excitation from incident waves, linear radiation from outgoing waves (generated by 

platform motion), and nonlinear effects. Depending on the code being used, each of these 

components may be treated in a different manner. Some codes may use a potential theory-based 

approach for modeling the hydrodynamics, some may use Morison’s equation, and some may 

use a combination of the two. Depending on which approach is used, only some of the sections 

in this chapter are applicable (see Table 4-1).

Table 4-: Applicable Sections for Hydrodynamics Modeling Approaches

Applicable Sections

Hydrodynamics 

modeling 

approach

Section 4.2 

Hydrostatics 

Section 4.3.2 

Potential flow 

theory

Section 4.3.3 

Morison’s Eq. 

Section 4.4 

Additional 

damping

Potential flow 

theory only X X X 

Morison’s 

equation only X X 

Potential theory 

and Morison’s 

equation 

X X X 

X = Needed if Morison-only model neglects buoyancy 

4.2 Hydrostatics

If using a code with either a potential-flow theory or Morison-based approach that neglects 

buoyancy, one must ensure that the hydrostatic component described in this section is included. 

The total load on the floating platform from linear hydrostatics, Hydrostatic Fi , is

( ) Hydrostatic Hydrostatic F q gV C q i = − ρ δ0 i3 ij j , (4-1) 

where ρ is the water density, g is the gravitational acceleration constant, V0 is the displaced 

volume of fluid when the platform is in its undisplaced position, δi3 is the (i,3) component of the 

Kronecker-Delta function (i.e. identity matrix), Hydrostatic Cij is the (i,j) component of the linear 

hydrostatic-restoring matrix from the effects of water-plane area at the center of buoyancy 

(COB), and j q is the j

th platform DOF. (Without the subscript, q represents the set of platform 

DOFs. Hydrostatic Fi depends on q as indicated.) In Eq. (4-1), subscripts i and j range from 1 to 6; 

one for each platform DOF (1 = surge, 2 = sway, 3 = heave, 4 = roll, 5 = pitch, 6 = yaw). 
12

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Einstein notation is used here, in which it is implied that when the same subscript appears in 

multiple variables in a single term, there is a sum of all of the possible terms. The loads are

positive in the direction of positive motion of DOF i. Equation (4-1) does not include the 

restoring effects of body weight (i.e. gravitational restoring). Instead, the gravitational restoring 

is assumed to be included within the structural dynamics models.

The first of the terms on the right-hand side of Eq. (4-1) represents the buoyancy force from 

Archimedes’ principle; that is, it is the force directed vertically upward and equal to the weight 

of the displaced fluid when the platform is in its undisplaced position. This term is nonzero only 

for the vertical heave-displacement DOF of the support platform (DOF i = 3) because the COB

lies on the centerline of the undeflected tower. The second of the terms on the right-hand side of 

Eq. (4-1) represents the change in the hydrostatic force and moment as the platform is displaced. 

The formulation for Hydrostatic Cij in terms of a platform’s water-plane shape, displaced volume, and 

COB location is given in (Jonkman 2007).

The water density is chosen to be 1,025 kg/m3

. From the external geometry of the floating 

platform, then, ρgV0 and Hydrostatic Cij were calculated to be:

ρgV0 =1.3989e8 N (4-2)

and

Hydrostatic

ij

00 0 0 0 0

00 0 0 0 0

0 0 3.836e6 N m 0 0 0

C

0 0 0 -3.776e8 Nm rad 0 0

0 0 0 0 -3.776e8 Nm rad 0

00 0 0 0 0

 

 

=

 

(4-3)

4.3 Hydrodynamics

The hydrodynamic loads―those associated with excitation from incident waves, radiation of 

outgoing waves from platform motion, added mass effects, and viscous forces―depend on a 

variety of factors, all of which will be discussed next. In this section, we demonstrate several 

methods to implement the hydrodynamic forces. The choice of method will depend on the 

hydrodynamics capabilities of the modeling tool being used. 

The scope of this document is focused on wave forces associated with radiation damping, addedmass, wave-excitation loads, and loads arising from flow separation. Hence, the second-order 

Quadratic Transfer Functions (QTFs) will not be defined here, but these effects could be 

considered in models that are capable. 

4.3.1 Dimensionless Numbers for Flow/Structure Interaction

For a floating platform interacting with surface waves, different formulations for the 

hydrodynamic loads apply to separated and nonseparated flows. For cylinders, the proper 
13

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formulation―and the hydrodynamic coefficients used within each formulation―depend, among 

other factors, on the Keulegan-Carpenter number, KC, and the oscillatory Reynolds number, Re, 

defined as (Gudmestad, et al 1998):

D

uT KC = (4-4)

and

ν

uD Re = (4-5)

where D is the cylinder diameter, ν is the kinematic viscosity of the fluid, T is the wave period, 

and u is the amplitude of the fluid velocity normal to the cylinder. 

The diameter to wavelength ratio, D/λ, is also an important factor that determines the proper 

formulation. For linear regular (i.e. single period) waves, the wavelength and (depth-dependent) 

wave velocity amplitude are related to the wave period and the wave height (i.e. twice the wave 

amplitude), H, by (Gudmestad, et al 1998):

[ ( )]

(kh)

k Z h

T

H

u

sinh

cosh + = π (4-6)

and

k

π λ 2 = (4-7)

where Z is the local depth (negative in value), h is the water depth (positive in value), and k is the 

wave number, which itself, is related to the wave period through the implicit dispersion 

relationship (Gudmestad, et al 1998):

( ) 2

2 4 tanh

gT

k kh π = . (4-8)

The water depth is taken to be 200 m. From the external geometry of the floating platform and 

the periodic sea states defined in Table 4-2 (from 1 = mild to 8 = extreme sea state), the 

Keulegan-Carpenter number and oscillatory Reynolds number are calculated. These values are 

presented as a function of depth (Z) along the semisubmersible for both the main turbine support 

column and the offset columns in Figure 4-1.
14 

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Table 4-: Periodic Sea State Definitions

Sea State T (s) H (m)

1 2.0 0.09 

2 4.8 0.67 

3 6.5 1.40 

4 8.1 2.44 

5 9.7 3.66 

6 11.3 5.49 

7 13.6 9.14 

8 17.0 15.24 

The upper and lower plots in Figure 4-1 show that, for the OC4-DeepCwind semisubmersible, 

the KC and oscillatory Re numbers grow with increasing severity in the wave conditions and 

decrease with depth (due to a decrease in the fluid velocity) for a constant-diameter section. The 

jump in values in the offset column plots occur at the point where the column diameter changes 

from that of the base column to the upper column. For the pontoons, the cylinder axis might not 

lie in the same plane as the two-dimensional flow; therefore, a range of values is possible based 

on the alignment of the flow direction with the cylinder axis. 

Flow separation becomes important when the KC number exceeds 2. For values lower than 2, 

potential-flow theory applies. Consequently, potential-flow theory applies for all of the large 

components of the semisubmersible in all but the most extreme wave conditions (sea states 6 – 

8), where separation will occur along the main column and upper portions of the offset columns. 

For the pontoons and cross members, flow separation occurs in sea states 5 – 8. 

Additionally, the diameter to wavelength ratio is calculated for the main and offset columns as 

well as the pontoons. This result is shown in Table 4-3. Table 4-3 shows that for the OC4-

DeepCwind semisubmersible, the diameter to wavelength ratio decreases with increasing 

severity of the wave conditions and is lowest for the smaller diameter members, such as the 

pontoons. Diffraction effects are important when this ratio exceeds 0.2 and are unimportant for 

smaller ratios. In Table 4-3, the cases where wave diffraction is important have been 

highlighted. These cases mainly include the lowest sea states, where the hydrodynamic loads are 

small anyway, but also include half of the sea state states for the large diameter base of the offset 

columns. 
15 

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Figure 4-: Dimensionless parameters for the OC4 DeepCwind semisubmersible

Table 4-: Ratio of Diameter/Wave Length 

Sea State 1 2 3 4 5 6 7 8 

Base 

Columns 3.844 0.667 0.364 0.234 0.163 0.120 0.083 0.054

Upper 

Columns 1.922 0.334 0.182 0.117 0.082 0.060 0.042 0.027 

Main 

Columns 1.041 0.181 0.099 0.063 0.044 0.033 0.023 0.015 

Pontoons 0.256 0.044 0.024 0.016 0.011 0.008 0.006 0.004 

0 2 4 6 8

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

KC number @ turbine support column

KC Number

Depth (m)

Sea State 1

Sea State 2

Sea State 3

Sea State 4

Sea State 5

Sea State 6

Sea State 7

Sea State 8

0 1 2 3 4 5

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

KC number @ Upper Column & Base

KC Number

Depth (m)

Sea State 1

Sea State 2

Sea State 3

Sea State 4

Sea State 5

Sea State 6

Sea State 7

Sea State 8

0 0.5 1 1.5 2

x 107

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Re number @ turbine support column

Re Number

Depth (m)

Sea State 1

Sea State 2

Sea State 3

Sea State 4

Sea State 5

Sea State 6

Sea State 7

Sea State 8

0 1 2 3 4 5 6

x 107

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Re number @ upper column & base

Re Number

Depth (m)

Sea State 1

Sea State 2

Sea State 3

Sea State 4

Sea State 5

Sea State 6

Sea State 7

Sea State 8
16

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4.3.2 Potential-Flow Theory

In view of the validity of potential-flow theory across many conditions, the linear potential-flow

problem was solved using WAMIT (Lee and Newman 2006), and the output from this analysis 

was shared with the participants. WAMIT uses a three-dimensional numerical-panel method in 

the frequency domain to solve the linearized potential-flow hydrodynamic radiation and

diffraction problems for the interaction of surface waves with offshore platforms of arbitrary 

geometry. The solution to the radiation problem, which considers the hydrodynamic loads on the 

platform associated with oscillation of the platform in its various modes of motion (which radiate 

outgoing waves), is given in terms of oscillation-frequency-dependent hydrodynamic addedmass and damping matrices, Aij and Bij respectively. The solution to the diffraction problem, 

which considers the hydrodynamic loads on the platform associated with excitation from incident 

waves, is given in terms of the wave-frequency and direction-dependent hydrodynamic waveexcitation vector, Xi. Whereas Aij and Bij are real-valued, Xi is complex-valued, with the 

magnitude determining the load normalized per unit wave amplitude and the phase determining 

the lag between the wave elevation and load. The subscripts here are consistent with those of 

Hydrostatic Cij discussed earlier and, as before, the loads are positive in the direction of positive 

motion. More information on potential-flow theory as it relates to floating platforms can be 

found in (Jonkman 2007). 

In WAMIT, the OC4-DeepCwind semisubmersible was modeled with one geometric plane of 

symmetry using a high-order representation of the geometry from a MultiSurf surface file. In the 

high-order representation of a WAMIT geometric description, the velocity potential on the body 

is represented by B-splines in a continuous manner. The model employed an average panel size 

of 2 m. To further improve the accuracy of the WAMIT results, options were selected to 

integrate the logarithmic singularity analytically, solve the linear system of equations using a 

direct solver, and remove the effects of irregular frequencies by manually paneling the free 

surface. These settings were beneficial because of the requirement of high-frequency output for 

time-domain analysis. The semisubmersible was analyzed in its undisplaced position (consistent 

with linear theory) and with finite water depth (200 m).

The magnitude and phase of the hydrodynamic wave-excitation vector from the linear diffraction 

problem are shown as a function of wave frequency in Figure 4-2 for incident waves propagating 

along the positive X-axis. For these waves, the loads in the direction of the sway (Mode 2), roll

(Mode 4), and yaw (Mode 6) DOFs are zero because the wave forces lie within the XZ platform 

plane.

The hydrodynamic added mass and damping matrices for the six DOFs are shown as a function 

of oscillation frequency in Figure 4-3. Only the upper triangular matrix elements are shown 

because the hydrodynamic added-mass and damping matrices are symmetric in the absence of 

forward motion. Also, because of the semisubmersible’s symmetries, the surge-surge elements 

of the frequency-dependent added-mass and damping matrices, A11 and B11, are identical to the 

sway-sway elements, A22 and B22. Likewise, the roll-roll elements, A44 and B44, are identical to 

the pitch-pitch elements, A55 and B55. This behavior exists because the OC4-DeepCwind 

semisubmersible has the same response at 0°, 120°, and 240° wave headings, (see Figure 4-4) 

which means that the X and Y responses must also be the same. Other matrix elements not 

shown are zero-valued. The zero and infinite-frequency limits of all elements of the damping 
17 

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matrix are zero, as required by potential theory, and peak at intermediate frequencies. The 

infinite-frequency added-mass matrix, A∞, is defined as:

2

2

2

6.49E+6 kg 0 0 0 -85.10E+6 kg m 0

0 6.49E+6 kg 0 85.10E+6 kg m 0 0

0 0 14.70E+6 kg 0 0 0

0 85.10E+6 kg m 0 7.21E+9 kg m 0 0

-85.10E+6 kg m 0 0 0 7.21E+9 kg m 0

0 0 0 0 0 4.87E+9 kg m

A∞

  ⋅

  ⋅

= ⋅ ⋅

⋅ ⋅

⋅  

Figure 4-: Hydrodynamic wave excitation per unit amplitude for the OC4 semisubmersible for 0° 

wave heading
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Figure 4-: Hydrodynamic added mass and damping for the OC4-DeepCwind semisubmersible
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Figure 4-: Simplified plan footprint of the OC4 semisubmersible

The linear memory effect is captured within time-domain hydrodynamics models through the 

time-convolution of the radiation impulse-response functions (i.e. the wave-radiation-retardation 

kernel), Kij, with the platform velocities. The memory effect captures the hydrodynamic load on 

the platform persisting from the outgoing free-surface waves (which induce a pressure field 

within the fluid domain) radiated away by platform motion. The radiation impulse-response 

functions can be found from the cosine transform of the frequency-dependent hydrodynamic 

damping matrix. The results of this computation, as performed within WAMIT’s frequency-totime (F2T) conversion utility, are shown in Figure 4-5. As before, only the upper triangular 

matrix elements of the symmetric Kij matrix are shown, and because of the semisubmersible’s

symmetry, the surge-surge elements, K11, are identical to the sway-sway elements, K22, and the 

roll-roll elements, K44, are identical to the pitch-pitch elements, K55. Most of the response and 

linear radiation damping decays to zero after about 40 seconds, even for the force-translation 

modes that may not be negligible. For more information on radiation theory, see (Jonkman 

2007). 

The second-order potential-flow solution, which includes mean-drift, slow-drift, and sumfrequency excitation, and higher-order solutions, were not solved. 
20 

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Figure 4-: Radiation impulse-response functions for the OC4-DeepCwind semisubmersible

4.3.3 Morison’s Equation 

In severe sea conditions, the hydrodynamic loads from linear potential-flow theory must be 

augmented with the loads brought about by flow separation. Moreover, many wind turbine 

dynamics codes cannot model hydrodynamics per linear potential-flow theory and use only a 

Morison-based approach for all sea conditions. To address these situations, a simplified 

hydrodynamics model using Morison’s formulation is presented next. 

The popular hydrodynamic formulation used in the analysis of fixed-bottom support structures 

for offshore wind turbines, Morison’s formulation, is applicable for calculating the 

hydrodynamic loads on cylindrical structures when (1) the effects of diffraction can be simplified 

with the long wavelength approximation, (2) radiation damping is negligible, and (3) flow 

separation may occur. The relative form of Morison’s equation accounts for wave loading from 

incident-wave-induced excitation, radiation-induced added mass, and flow-separation-induced 

viscous drag. In the next two sections we review the representation of Morison’s equation in 

flow transverse to the structure and for a heave plate.
21

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4.3.3.1 Transverse Flow

For a cylinder in steady, transverse flow, the wave forces per unit length on the cylinder with 

velocity q can be expressed as:

2 2 1 ( ) (1 ) 2 4 4 d a a

D D F C Du q u q C u C q π π

= − −++ − ρ ρρ     (4-9)

where ρ is the fluid density, D is the diameter of the cylinder, Cd is the drag coefficient, and Ca is 

the added mass coefficient. Equation (4-9) is comprised of a quadratic drag term, the fluid-inertia

excitation force, and the added-mass term. In addition, some Morison-based software programs 

will account for the dynamic pressure caused by the varying water elevation.

From the discussion in Section 4.3.1, Morison’s equation is a reasonable approximation for the 

OC4-DeepCwind semisubmersible in most wave conditions because (1) diffraction effects can 

be approximated by long wavelength theory in moderate to severe sea states, (2) radiation 

damping in most modes is small, and (3) flow separation will occur in severe sea states along the 

upper column regions of the semisubmersible. In the next sections, we demonstrate the 

methodology to obtain the drag coefficient Cd and added-mass coefficient Ca in Equation (4-9).

Added Mass

The added-mass coefficient Ca to be used in Equation (4-9) was selected such that CaρV equaled

the zero-frequency limit of A11 in the surge direction from the potential-flow solution (Figure 

4-3). This equivalency was based on the assumption that Ca is independent of water depth and 

the motion is of low frequency. By this equivalency, an initial estimate for the added mass 

coefficient for each member of the DeepCwind semisubmersible was 0.63. This value should be 

used for all members in all platform degrees of freedom, with the exception of heave, which will 

be addressed below.

To verify the cross-flow added mass coefficient Ca = 0.63, three models were assembled in 

OrcaFlex (a commercial modeling tool for offshore structures) using the platform geometric 

properties outlined in Table 3-1 through Table 3-3, each with different representations of the 

added mass:

• Model 1: models the added mass force contribution using Morison’s equation

• Model 2: includes only the WAMIT added mass coefficients 

• Model 3: no added mass is modeled 

Model 3 was constructed purely for curiosity measures to quantify the significance of added 

mass on the system. The comparison was performed in OrcaFlex because this software allows 

for input of the WAMIT solution and provides an opportunity to attach Morison added-mass 

elements discretely along the various column members and cross braces, thus allowing each 

discrete component to be modeled with unique hydrodynamic properties. Each model was 

prescribed an impulse force to compel the platform to freely oscillate. By comparing the motion 

of Model 1 with Model 2, it was decided that Ca = 0.63 is a satisfactory coefficient to use when 

modeling the cross-flow column added mass using Morison’s equation.
22 

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Viscous Drag

With the added mass coefficient, Ca, selected, we can now focus on determining the drag 

coefficient Cd for Equation (4-9). For a cylinder in steady, transverse flow, the drag coefficient 

is determined based on the Reynold’s number using the relationship provided in Figure 4-6

(Catalano, et al 2003). In this figure, the Re values for each of the semisubmersible’s members 

in various sea states are plotted using markers. As shown, the drag force can vary greatly across 

the flow regimes the system is likely to encounter, and therefore an average Cd value is 

calculated for each member, as outlined in Appendix I and summarized in Table 4-4. Note that 

these values are derived for the full-scale system and associated Re values. A different set of 

Cd values would be chosen to match the drag characteristics of the DeepCwind tank tests, which 

were performed at model scale and therefore encompassed very different Re values. The KC 

number was not considered in this calculation. 

Figure 4-: Distribution of the drag coefficient as a function of the Re number

Table 4-: Mean Cd Values Across All Sea States for the Full-Scale OC4 Semisubmersible 

Column D=1.6 m D=6.5 m D=12 m D=24 m

Cd 0.63 0.56 0.61 0.68 

4.3.3.2 Heave Plate

Unlike a cylinder in cross-flow, the force on a heave plate does not scale proportionally to the 

displaced fluid (because the heave plate displaces little volume). The three base columns are 
23

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considered as heave plates. The hydrodynamic heave force for a heave plate is formulated using 

this modified Morison’s equation (to include dynamic pressure) as given below:

Fz = 1

2 ρCdzAC|w − q̇

3|(w − q̇3) + ρCazVR(ẇ − q̈3) +

π

4

Dh

2pb − π

4 (Dh

2 − Dc

2)pt (4-10)

where Cdz is the drag coefficient in the heave direction, AC is the cross-sectional area of the 

heave-plate in the Z-direction, w is the vertical wave particle velocity, q̇

3 is the heave velocity of 

the heave-plate, Caz is the added mass coefficient in the heave direction, VR is the reference 

volume, ẇ is the vertical wave particle acceleration, q̈

3 is the heave acceleration of the heaveplate, Dh is the diameter of the heave-plate, Dc is the diameter of the upper column (which is 

placed on top of the heave-plate), and pb and pt are the dynamic pressure acting at the bottom 

and top faces of the heave-plate.

In equation (4-10), the first term corresponds to the drag force in the heave direction, the second 

term corresponds to the combined fluid acceleration (scattering) and added mass force, and the 

last two terms correspond to the Froude-Krylov force expressed in terms of dynamic pressure.

Please note that the viscous drag, fluid-acceleration, and added mass terms are shown for an 

entire heave-plate collectively, but could also be expressed with separate forces on the bottom 

and top faces.

Added Mass

A separate added mass coefficient for the heave direction must be used to properly model the 

column-base added mass. The following relationship can be used to find the added mass 

coefficient, Caz, in the heave direction:

33 (0) az

R

A C

ρV = (4-11)

where A33(0) is the zero-frequency limit added mass in the heave direction obtained from 

WAMIT (which can be found in Figure 4-3). Equation (4-11) is valid for the entire structure. To 

consider only one of the three columns, the value for A33(0) would need to be divided by three,

and the reference volume would only be for that one column rather than the entire structure. This 

assumes that the contribution of the cross braces and main column to the heave added mass is 

negligible. One can use any appropriate combination of Caz and VR that produces the correct 

added mass. 

For example, if Caz is chosen as 1.0, then for one column: VR = A33(0)/3

ρ = 4.88E+3 m

3

. 

Similarly, if VR is chosen as VR = 4

3

πr3, where r is the heave plate radius, then Caz = A33(0)/3

4

3

πρr3 = 

0.67 for one column.

Viscous Drag

As was true for the added-mass coefficient, the heave plates require a separate drag coefficient 

for the heave direction. Assuming the heave plate emulates a flat plate with flow normal to the 

face, a value of Cdz = 4.8 is selected. This value was found by matching a coupled 
24 

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FAST+OrcaFlex simulation with DeepCwind tank-test data in the heave direction. In this 

situation—due to sharp corners—the drag coefficient is no longer strongly dependent on Re

number and will be the same for a full-scale or model-scale system, which allows us to derive the 

value for our full-scale system through comparison to the tank-test data. 

4.3.4 Quadratic Drag for Other Models

For codes that model potential flow without Morison elements, extra damping is needed to 

accurately represent the damping in a real system. To determine the amount of additional 

damping required, free-decay simulations of the semisubmersible using linear potential-flow 

theory were executed then compared to simulations using a combination of potential-flow theory 

and the viscous term in Morison’s equation. Each model was constructed in FAST and, as 

needed, coupled to OrcaFlex (Jonkman and Buhl 2005). For the models not representing viscous 

elements discretely, i.e. those omitting the viscous component in Equation (4-9), additional 

quadratic drag was added to the model on top of the potential-flow contribution. This drag is 

implemented according to the following equation: 

( ) Additional Drag quad F q B qq i = − ij j j   , (4-12) 

where quad Bij is the (i,j) component of the additional quadratic drag matrix, and j q is the first 

time derivative of the j

th platform DOF. 

The quantities labeled as quad Bii in Equation (4-12) represent the required quadratic viscous drag 

coefficient needed to match a simulation using Equation (4-12) against a simulation using 

discrete Morison viscous drag elements. The off-diagonal elements of matrix quad Bij are 

neglected. 

Table 4-: Quadratic Drag Coefficients for the FAST Model 

Surge

Ns2/m2

Sway

Ns2/m2

Heave

Ns2/m2

Roll

Nms2/rad2

Pitch

Nms2/rad2

Yaw

Nms2/rad2

quad Bii

* 3.95E+5 3.95E+5 3.88E+6 3.70E+10 3.70E+10 4.08E+9 
25 

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4.4 Summary

Table 4-6 summarizes the hydrodynamic properties (except the linear potential-flow solution) 

that were discussed in this section.

Table 4-: Floating Platform Hydrodynamic Properties

Water density (ρ) 1025 kg/m3

Water depth (h) 200 m

Displaced water in undisplaced position (V0) 13917 m3

Center of buoyancy below SWL 13.15 m

Buoyancy force in undisplaced position (ρgV0) 1.3989E+8 N 

Hydrostatic restoring in heave (C33

Hydrostatic) 3.836E+06 N/m 

Hydrostatic restoring in roll (C44

Hydrostatic) -3.776E+08 N-m/rad

Hydrostatic restoring in pitch (C55

Hydrostatic) -3.776E+08 N-m/rad

Added-mass coefficient (Ca ) for all members 0.63 

Added-mass coefficient (Caz ) for base column in zdirection1.0 

Drag coefficient (Cd ) for main column 0.56 

Drag coefficient (Cd ) for upper columns 0.61 

Drag coefficient (Cd ) for base columns 0.68 

Drag coefficient (Cd ) for pontoons and cross members 0.63 

Drag coefficient (Cdz ) for base columns in z-direction 4.8 

Additional quadratic drag in surge (B11

quad) 3.95E+5 Ns2

/m2

Additional quadratic drag in sway (B22

quad) 3.95E+5 Ns2

/m2

Additional quadratic drag in heave (B33

quad) 3.88E+6 Ns2

/m2

Additional quadratic drag in roll (B44

quad) 3.70E+10 Nms2

/rad2

Additional quadratic drag in pitch (B55

quad) 3.70E+10 Nms2

/rad2

Additional quadratic drag in yaw (B66

quad) 4.08E+9 Nms2

/rad2
26 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

5 Mooring System Properties

5.1 Overview

To secure the platform, the OC4-DeepCwind semisubmersible is moored with three catenary 

lines spread symmetrically about the platform Z-axis. The mooring layout in the basin is shown 

in Figure 5-1. The mooring line properties are listed in Table 5-1. 

The fairleads (body-fixed locations where the mooring lines attach to the platform) are located at 

the top of the base columns at a depth of 14.0 m below the SWL and at a radius of 40.87 m from 

the platform centerline. The anchors (fixed to the inertia frame) are located at a water depth of 

200 m below the SWL and at a radius of 837.6 m from the platform centerline. One of the lines 

is directed along the negative X-axis (in the XZ-plane). The two remaining lines are distributed 

uniformly around the platform, such that each line, fairlead and anchor, is 120o apart when 

looking from above, as shown in Figure 5-1. These properties are relative to the undisplaced 

position of the platform. Each of the three lines has an unstretched length of 835.5 m, a diameter 

of 0.0766 m, an equivalent mass per unit length of 113.35 kg/m, an equivalent apparent mass in 

fluid per unit length of 108.63 kg/m, and an equivalent extensional stiffness of 753.6 MN. Drag 

and added mass of the mooring lines due both to hydrodynamic loading and seabed interaction 

will be modeled using the coefficients provided in Table 5-1, which summarizes all of the 

mooring properties. In addition, the structural damping of the mooring line is set to 2% stiffness 

proportional damping. 

Table 5-1: Mooring System Properties 

Number of Mooring Lines 3

Angle Between Adjacent Lines 1200

Depth to Anchors Below SWL 200 m

Depth to Fairleads Below SWL 14 m

Radius to Anchors from Platform Centerline 837.6 m

Radius to Fairleads from Platform Centerline 40.868 m

Unstretched Mooring Line Length 835.5 m

Mooring Line Diameter 0.0766 m

Equivalent Mooring Line Mass Density 113.35 kg/m 

Equivalent Mooring Line Mass in Water 108.63 kg/m 

Equivalent Mooring Line Extensional Stiffness 753.6 MN 

Hydrodynamic Drag Coefficient for Mooring Lines 1.1 

Hydrodynamic Added-Mass Coefficient for Mooring Lines 1.0 

Seabed Drag Coefficient For Mooring Lines 1.0 

Structural Damping of Mooring Lines 2.0% 

For the codes that cannot model individual mooring lines directly, three simplified models are 

presented in the following sections. The first is a linearized model of the complete mooring 

system. The second is a nonlinear model of the complete mooring system. The third is a 

nonlinear model of an individual mooring line. 
27 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 5-1: Mooring line arrangement

5.2 Linearized Mooring Model

If the mooring system compliance were inherently linear and mooring inertia and damping were 

ignored, as assumed in the first simplified model, the total load on the support platform from the 

contribution of all mooring lines, Lines Fi , would be 

( ) Lines Lines,0 Lines F q =F C q i i − ij j , (5-13) 

where Lines,0 Fi is the i

th component of the total mooring system load acting on the support 

platform in its undisplaced position, Lines Cij is the (i,j) component of the linearized restoring 

matrix from all mooring lines, and j q is the j

th platform DOF. (Without the subscript, q

represents the set of platform DOFs. Lines Fi depends on q as indicated.) For catenary mooring 

lines, Lines,0 Fi represents the pre-tension at the fairleads from the weight of the mooring lines not 

resting on the seafloor in water. Lines Cij is the combined result of the elastic stiffness of the 

mooring lines and the effective geometric stiffness brought about by the weight of the lines in 

water, depending on the layout of the mooring system. The loads are positive in the direction of 

positive motion of DOF i. 

Waves/

Wind

 00
28

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For the mooring system considered here, Lines,0 Fi and Lines Cij were calculated by performing a 

linearization analysis in FAST (Jonkman and Buhl 2005) about the undisplaced position of the 

platform (i.e. about the linearization point where all DOF displacements are zero-valued). 

(FAST includes a mooring system model [Jonkman 2007] that was needed to make these 

calculations.) The linearization analysis involves independently perturbing the platform DOFs 

and measuring the resulting variations in mooring loads. Within FAST, the partial derivatives 

are computed using the central-difference-perturbation numerical technique. The results are as 

follows:

Lines,0

i

0

0

1,839,000 N F

0

0

0

 

 

 

  − =  

 

 

 

 

(5-14)

and

Lines

ij

7.08e4 N m 0 0 0 1.08e5 N rad 0

0 7.08e4 N m 0 1.08e5 N rad 0 0

0 0 1.91e4 N m 0 0 0

C

0 1.07e5 Nm m 0 8.73e7 Nm rad 0 0

1.07e5 Nm m 0 0 0 8.73e7 Nm rad 0

0 0 0 0 0 1.17e8 Nm rad

  −

 

=

−

 

.(5-15)

5.3 Nonlinear Mooring Model

The linear model is only valid for small displacements about the linearization point. For larger 

displacements, it is important to capture the nonlinear relationships between load and 

displacement, as provided in the second simplified model. In general, all six components of 

Lines F depend nonlinearly on all six displacements of q. (Without the subscript, Lines F

represents the set of mooring-system loads that include three forces and three moments.) For the 

mooring system considered here, these load-displacement relationships were found numerically 

using FAST by considering discrete combinations of the displacements. The surge and sway 

displacements (q1 and q2) were varied from -32 to 32 m in steps of 8 m. The heave displacement 

(q3) was varied from -9 to 9 m in steps of 3 m. The roll and pitch displacements (q4 and q5) were 

varied from -13o to 13o in steps of 3.25o. The yaw displacement (q6) was varied from -27o to 27o 

in steps of 6.75o. All six components of Lines F were calculated for every combination of these 

displacements, for a total of (9x9x7x9x9x9 =) 413,343 discrete combinations. The upper and 

lower bounds in these variations were determined by examining the extreme responses of the 

DeepCwind semisubmersible system under wind and wave excitation estimated prior to the tank 

tests in (Robertson and Jonkman 2011). The OC4 DeepCwind semisubmersible model is slightly 

different from the model in this analysis, but the dynamics should largely be the same. The step 

sizes were chosen to produce reasonable resolution in the nonlinear response at a minimal 

computational cost. All of the data—that is, all six components of Lines F dependent on all six 

displacements of q—were written to a text file: “MooringSystemFD.txt.”
29

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 5-2 shows the load-displacement relationships for the DeepCwind semisubmersible

mooring system when each platform DOF is varied independently with all other displacements 

having a zero value (i.e. Figure 5-2 presents a sample of one-dimensional [1D] loaddisplacement relationships). The relationships include some interesting asymmetries that result 

from the nonlinear behavior of the three-point mooring system. Whereas the loads are either 

symmetric or antisymmetric about zero for the sway (q2) and roll (q4) displacements, the loads 

are asymmetric about zero for the surge (q1) and pitch (q5) displacements. For the surge and 

pitch displacements, the mooring system stiffens up—and the surge forces, heave forces, and 

pitching moments increase nonlinearly—when the fairleads translate along the X-axis. This 

asymmetry also induces surge forces and pitching moments when the fairleads translate along 

the Y-axis caused by sway and roll displacements. Also, the heave forces change with all 

displacements because these displacements cause more line to lift off of—or allow more line to 

settle on—the seabed. The slopes of these load-displacement relationships about zerodisplacement are consistent with the elements of the linearized restoring matrix, Lines Cij , presented 

above.

Figure 5-3 shows the nonlinear relationships for the surge forces and pitching moments 

associated with surge and pitch displacements and the sway forces and roll moments associated 

with sway and roll displacements with all other displacements having a zero value (i.e. Figure 

5-2 presents a sample of two-dimensional [2D] load-displacement relationships). The 

combinations of surge and pitch displacements and sway and roll displacements were plotted 

because those directions are coupled. As in Figure 5-2, Figure 5-3 also shows the asymmetry 

about zero-displacement between translations of the fairlead along the X- and Y-axes (caused by

surge and pitch displacements and sway and roll displacements, respectively). The data in 

Figure 5-3 are also consistent with the results of Figure 5-2, as some of the results of Figure 5-2 

are slices through the data of Figure 5-3 when one of the displacements in Figure 5-3 has a zero

value.
30 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 5-2: Load-displacement relationships for the OC3-Hywind mooring system in 1D

-30 -20 -10 0 10 20 30

-1.5

-1

-0.5

0

0.5

1

1.5

x 104

Displacement in Surge (m)

Restoring Force (F1-F3), kN

Restoring Force/Moment vs. Displacement in Surge

-30 -20 -10 0 10 20 30

-6

-4

-2

0

2

4

6

x 104

-30 -20 -10 0 10 20 30

-6

-4

-2

0

2

4

6

x 104

-30 -20 -10 0 10 20 30

-6

-4

-2

0

2

4

6

x 104Restoring Moment (F4-F6), kN m⋅

Surge

Roll

Sway

Pitch

Heave

Yaw

-10 -5 0 5 10

-2000

-1000

0

1000

2000

Displacement in Roll (deg)

Restoring Force (F1-F3), kN

Restoring Force/Moment vs. Displacement in Roll

-10 -5 0 5 10 -2

-1

0

1

2

x 104

-10 -5 0 5 10 -2

-1

0

1

2

x 104

-10 -5 0 5 10 -2

-1

0

1

2

x 104Restoring Moment (F4-F6), kN m⋅

Surge

Roll

Sway

Pitch

Heave

Yaw

-30 -20 -10 0 10 20 30 -1

-0.5

0

0.5

1

x 104

Displacement in Sway (m)

Restoring Force (F1-F3), kN

Restoring Force/Moment vs. Displacement in Sway

-30 -20 -10 0 10 20 30 -4

-2

0

2

4

x 104

-30 -20 -10 0 10 20 30 -4

-2

0

2

4

x 104

-30 -20 -10 0 10 20 30 -4

-2

0

2

4

x 104Restoring Moment (F4-F6), kN m⋅

Surge

Roll

Sway

Pitch

Heave

Yaw

-10 -5 0 5 10

-2000

-1000

0

1000

2000

Displacement in Pitch (deg)

Restoring Force (F1-F3), kN

Restoring Force/Moment vs. Displacement in Pitch

-10 -5 0 5 10 -2

-1

0

1

2

x 104

-10 -5 0 5 10 -2

-1

0

1

2

x 104

-10 -5 0 5 10 -2

-1

0

1

2

x 104Restoring Moment (F4-F6), kN m⋅

Surge

Roll

Sway

Pitch

Heave

Yaw

-8 -6 -4 -2 0 2 4 6 8

-4000

-2000

0

2000

4000

Displacement in Heave (m)

Restoring Force (F1-F3), kN

Restoring Force/Moment vs. Displacement in Heave

-8 -6 -4 -2 0 2 4 6 8 -4

-2

0

2

4

x 104

-8 -6 -4 -2 0 2 4 6 8 -4

-2

0

2

4

x 104

-8 -6 -4 -2 0 2 4 6 8 -4

-2

0

2

4

x 104Restoring Moment (F4-F6), kN m⋅

Surge

Roll

Sway

Pitch

Heave

Yaw

-20 -10 0 10 20 -4000

-2000

0

2000

4000

Displacement in Yaw (deg)

Restoring Force (F1-F3), kN

Restoring Force/Moment vs. Displacement in Yaw

-20 -10 0 10 20 -1

-0.5

0

0.5

1

x 105

-20 -10 0 10 20 -1

-0.5

0

0.5

1

x 105

-20 -10 0 10 20 -1

-0.5

0

0.5

1

x 105

Restoring Moment (F4-F6), kN m⋅

Surge

Roll

Sway

Pitch

Heave

Yaw
31 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 5-3: Load-displacement relationships for the OC4 DeepCwind semisubmersible mooring 

system in 2D. 

5.4 Nonlinear Mooring Model with One Line

The two simplified mooring system models presented previously only capture the total load from 

the complete mooring system acting on the support platform; no information is given regarding 

the reactions within each individual mooring line. The third simplified mooring system model 

addresses this limitation by giving the load-displacement relationship of an individual mooring 

line. The model is presented in the output format of the MIMOSA mooring analysis software 

(Det Norske Veritas), which gives the qausi-static reactions of an individual mooring line as a 

function of the horizontal distance between the fairlead and anchor (see Figure 5-4). The vertical 

distance between the fairlead and anchor is fixed at ( 200 m – 14 m ) = 186 m in this model; 186 

m is equal to the vertical distance between the fairlead and anchor when the platform is 

undisplaced in pitch, roll, and heave. In the nomenclature of MIMOSA format, “DISTANCE” is 

the horizontal distance between the fairlead and anchor. “TENSION” is the total vector 

combination of the horizontal and vertical tensions in the mooring line at the fairlead. 

“H.TENSION” is the horizontal component of this fairlead tension. “SUSPL” is the distance

along the line from the fairlead to the equivalent point on the unstretched mooring line where the 
32 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

point on the stretched mooring line first touches the seabed. “TEN.ANCH” is the total vector 

combination of the horizontal and vertical tensions in the mooring line at the anchor. 

For the OC4 DeepCwind semisubmersible mooring lines, this model was derived numerically 

using FAST by considering a range of discrete horizontal distances for one of the three identical 

mooring lines. That is, DISTANCE was varied from 649 m to 902.5 m in steps of 0.5 m. The 

upper and lower bounds were determined by choosing limits far beyond the likely distances that 

will be covered when the full system model is run in time-domain simulations. The step size was 

chosen so as to produce reasonable resolution in the nonlinear response at a minimal 

computational cost. All of the data—that is, TENSION, H.TENSION, SUSPL, and TEN.ANCH 

dependent on DISTANCE—were written to a text file: “MooringLineFD.txt.” 

Figure 5-5 shows the data. When DISTANCE equals 649 m, H.TENSION equals zero, and the 

tension is only in the vertical direction. When DISTANCE is above 810 m, the SUSPL is 

constantly equal to 835.5 m, the unstretched length of the mooring line, which means that no 

portion of the mooring line is on the seabed. Below 810 m of DISTANCE, a portion of the 

mooring line rests on the seabed, such that SUSPL is less than the total unstretched length of the 

mooring line. TENSION is larger than H.TENSION because it includes a vertical component 

combined from the elastic stretching and the weight in water of the mooring line not resting on 

the seabed. For DISTANCE larger than 810 m, no portion of the mooring line rests on the 

seabed, such that SUSPL equals 835.5 m and the anchor tension includes a nonzero vertical 

component that causes TEN.ANCH to exceed the value of H.TENSION. All tensions rise 

dramatically as the mooring line gets more and more taut.

Figure 5-4: Mooring line in a local coordinate system 

TEN.ANCH

SUSPL

�TENSION2– H. TENSION2

DISTANCE

H.TENSION

186 m

TENSION
33 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

 

Figure 5-5: Load-displacement relationships for one mooring line
34 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

6 Control System Properties

The NREL 5-MW wind turbine uses a conventional variable-speed, variable blade-pitch-tofeather control system (Jonkman, et al 2009). A consequence of conventional pitch-to-feather 

control of wind turbines, though, is that steady-state rotor thrust is reduced with increasing wind 

speed above rated. As pointed out by (Nielsen, et al 2006, p. 673), “this effect may introduce 

negative damping in the system that may lead to large resonant motions of [a] floating wind 

turbine.” As the analyses (Jonkman 2009, Larsen and Hanson 2007, Nielsen et al 2006, and 

Skaare et al 2007) have demonstrated, it is important that the damping of the platform-pitch 

mode be positive and kept as large as possible. 

Modifications were made to the original control system for the NREL 5-MW turbine for the 

OC3-Hywind project to avoid the negative damping issue. These modifications included a 

reduction of gains in the blade-pitch-to-feather control system and a change in the generatortorque control strategy when operating at rated power (i.e., Region 3 control). For this first 

modification, the gains in the blade-pitch-to feather control system were reduced so that the 

blade-pitching control frequency was below that of the dominant frequencies of the system, the 

most pertinent being the platform-pitch natural frequency. The second modification was to 

change the control law in Region 3 from constant generator power to constant generator torque, 

which will reduce the rotor-speed excursions that are exaggerated by the reduction in gains in the 

blade-pitch controller.

The platform-pitch natural frequency of the OC4-DeepCwind semisubmersible system (0.037 

Hz) is very similar to that of the OC3-Hywind spar (0.034 Hz). Because the platform-pitch 

frequency is used to set the gains of the blade-pitch-to-feather control system, it was decided to 

re-use the control methodology developed for the OC3-Hywind spar for the semisubmersible, 

including the change in Region 3 from constant power to constant generator torque. This 

control methodology was incorporated into a control system dynamic link library (DLL), which

is in the style of Garrad Hassan’s BLADED wind turbine software package (Bossanyi 2003). 

Table 6-1 summarizes the updated properties of the baseline control system discussed in this 

section.

Table 6-1: Baseline Control System Property Modifications

Proportional Gain at Minimum Blade-Pitch Setting 0.006275604 s 

Integral Gain at Minimum Blade-Pitch Setting 0.0008965149 

Constant (Rated) Generator Torque in Region 3 43,093.55 N-m 
35

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

References

Abbott, M.B. and Price, W.A., 1993. Coastal, Estuarial and Harbour Engineers’ Reference 

Book. Chapman & Hall, London, UK. 

Bossanyi, E. A., GH Bladed Version 3.6 User Manual, 282/BR/010, Bristol, UK: Garrad Hassan 

and Partners Limited, December 2003.

Catalano, P.; Wang, M.; Iaccarino, G.; Moin, P. 2003. “Numerical simulation of the flow around 

a circular cylinder at high Reynolds numbers.” International Journal of Heat and Fluid Flow, 

24, pp. 463-469.

Det Norske Veritas. MIMOSA Mooring Analysis Software.

Goupee, A.J.; Koo, B.J.; Lamrakos, K.F.; Kimball, R., “Offshore Wind Energy: Model Tests for 

Three Floating Wind Turbine Concepts,” Proceedings of the Offshore Technical Conference, 

April 30 – May 3, 2012, Houston, TX.

Gudmestad, O. T., et al, Basics of Offshore Petroleum Engineering and Development of Marine 

Facilities — With Emphasis on the Arctic Offshore, Stavanger, Moscow, St. Petersburg, 

Trondheim, December 1998.

Jonkman, J. M. and Buhl Jr., M. L. FAST User’s Guide, NREL/EL-500-38230 (previously 

NREL/EL-500-29798), Golden, CO: National Renewable Energy Laboratory, August 2005.

Jonkman, J. M., Dynamics Modeling and Loads Analysis of an Offshore Floating Wind Turbine, 

Ph.D. Thesis, Department of Aerospace Engineering Sciences, University of Colorado, Boulder, 

CO, 2007; NREL/TP-500-41958, Golden, CO: National Renewable Energy Laboratory.

Jonkman, J.; Butterfield, S.; Musial, W. and Scott, G., Definition of a 5-MW Reference Wind 

Turbine for Offshore System Development, NREL/TP-500-38060, Golden, CO: National 

Renewable Energy Laboratory, February 2009.

Jonkman, J., Definition of the Floating System for Phase IV of OC3, NREL/TP-500-47535, 

Golden, CO: National Renewable Energy Laboratory, May 2010.

Kooijman, H. J. T.; Lindenburg, C.; Winkelaar, D., and van der Hooft, E. L., “DOWEC 6 MW 

Pre-Design: Aero-elastic modeling of the DOWEC 6 MW pre-design in PHATAS,” DOWEC 

Dutch Offshore Wind Energy Converter 1997–2003 Public Reports [CD-ROM], DOWEC 

10046_009, ECN-CX--01-135, Petten, the Netherlands: Energy Research Center of the 

Netherlands, September 2003.

Larsen, T. J. and Hanson, T. D., “A Method to Avoid Negative Damped Low Frequent Tower 

Vibrations for a Floating, Pitch Controlled Wind Turbine,” Journal of Physics: Conference 

Series, The Second Conference on The Science of Making Torque From Wind, Copenhagen, 

Denmark, 28–31 August 2007, [online journal], Vol. 75, 2007, 012073, URL: 

http://www.iop.org/EJ/article/1742-6596/75/1/012073/jpconf7_75_012073.pdf?requestid=SpEuhRBu3BG0xV3r2wi7Kg, [cited 28 August 2007].
36

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Lee, C. H. and Newman, J. N., WAMIT® User Manual, Versions 6.3, 6.3PC, 6.3S, 6.3S-PC, 

Chestnut Hill, MA: WAMIT, Inc., 2006.

Nielsen, F. G.; Hanson, T. D. and Skaare, B., “Integrated Dynamic Analysis of Floating Offshore 

Wind Turbines,” Proceedings of OMAE2006 25th International Conference on Offshore 

Mechanics and Arctic Engineering, 4–9 June 2006, Hamburg, Germany [CD-ROM], Houston, 

TX: The American Society of Mechanical Engineers (ASME International) Ocean, Offshore and 

Arctic Engineering (OOAE) Division, June 2006, OMAE2006-92291.

Robertson, A.N. and Jonkman, J., “Loads Analysis of Several Offshore Floating Wind Turbine 

Concepts,” Proceedings of the 21st International Offshore and Polar Engineering Conference, 

June 2011.

Skaare, B.; Hanson, T. D. and Nielsen, F. G., “Importance of Control Strategies on Fatigue Life 

of Floating Wind Turbines,” Proceedings of OMAE2007 26th International Conference on 

Offshore Mechanics and Arctic Engineering, 10–15 June 2007, San Diego, CA [CD-ROM], 

Houston, TX: The American Society of Mechanical Engineers (ASME International) Ocean, 

Offshore and Arctic Engineering (OOAE) Division, June 2007, OMAE2007-29277.
37

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Appendix I: Calculation of Average Cd Value

Figure A1-1 shows the distribution of the drag coefficient as a function of the sea state at full 

scale. This figure gives the drag coefficient range for all column diameters being considered on 

the OC4-DeepCwind semisubmersible. To arrive at the Re number, an average of the wave 

particle velocity is calculated by integrating Eq. (4-6) along the submerged column depth: 

upper

lower

upper

lower

sinh( )

sinh( )

sinh( ) ( )

1 cosh( )

upper lower upper lower

Z l

Z l

l

l kh

kZ kh

l l Tk

H dZ

kh

kZ kh

T

H

l l

V

=

=

+

− = +

− = ∫

π π (A1-16)

where lupper and llower define the upper and lower (respectively) limits of the column as measured 

below the still water line. 

The resulting value for Cd is calculated by interpolating experimental data found in [Catalano, et 

al 2003]. When contrasted against experimental data, it becomes apparent the Re number for 

each column member spans the range Re = 105 – 107

. The bulk of the Cd values within this range 

are between 0.3 and 1.2.

To assist in the selection of a single drag coefficient value for a particular column, the values for 

Cd are plotted as a function of depth in Figure AII-1. Each figure illustrates how the drag 

coefficient varies between sea-state 2 and sea-state 8, as well as the in-between sea-states. The 

blue ticks in Figure AII-1 mark the Cd values for the sea-states between 2 and 8. Sea-state 1 is 

not shown in these figure because their value for Re is low, hence skewing the results. From 

here, the depth-averaged drag coefficient can be formed. 
38 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure AI-1: Variation of the drag coefficient across all sea-states. The mean value for Cd at 

incremental depths is also registered in the figures.